Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #33 Nov 08 2022 08:07:58
%S 1,16383,2391484,134209536,1525878906,39179682372,113037178808,
%T 1099444518912,3812797945332,24998474116998,37974983358324,
%U 320959957991424,328114698808274,1851888100411464,3649114989636504
%N a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 15.
%C a(n) is the number of lattices L in Z^14 such that the quotient group Z^14 / L is C_n. - _Álvar Ibeas_, Nov 26 2015
%H Enrique Pérez Herrero, <a href="/A161025/b161025.txt">Table of n, a(n) for n = 1..5000</a>
%H Jin Ho Kwak and Jaeun Lee, <a href="https://doi.org/10.1142/9789812799890_0005">Enumeration of graph coverings, surface branched coverings and related group theory</a>, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%H <a href="/index/J#nome">Index to Jordan function ratios J_k/J_1</a>.
%F a(n) = J_14(n)/J_1(n) where J_14 and J_1(n) = A000010(n) are Jordan functions. - _R. J. Mathar_, Jul 12 2011
%F From _Álvar Ibeas_, Nov 26 2015: (Start)
%F Multiplicative with a(p^e) = p^(13e-13) * (p^14-1) / (p-1).
%F For squarefree n, a(n) = A000203(n^13). (End)
%F From _Amiram Eldar_, Nov 08 2022: (Start)
%F Sum_{k=1..n} a(k) ~ c * n^14, where c = (1/14) * Product_{p prime} (1 + (p^13-1)/((p-1)*p^14)) = 0.1388226555... .
%F Sum_{k>=1} 1/a(k) = zeta(13)*zeta(14) * Product_{p prime} (1 - 2/p^14 + 1/p^27) = 1.00006146517418... . (End)
%p A161025 := proc(n)
%p add(numtheory[mobius](n/d)*d^14,d=numtheory[divisors](n)) ;
%p %/numtheory[phi](n) ;
%p end proc:
%p for n from 1 to 5000 do
%p printf("%d %d\n",n,A161025(n)) ;
%p end do: # _R. J. Mathar_, Mar 15 2016
%t A161025[n_]:=DivisorSum[n,MoebiusMu[n/#]*#^(15-1)/EulerPhi[n]&]
%t f[p_, e_] := p^(13*e - 13) * (p^14-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* _Amiram Eldar_, Nov 08 2022 *)
%o (PARI) vector(100, n, sumdiv(n^13, d, if(ispower(d, 14), moebius(sqrtnint(d, 14))*sigma(n^13/d), 0))) \\ _Altug Alkan_, Nov 26 2015
%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^14 - 1)*f[i,1]^(13*f[i,2] - 13)/(f[i,1] - 1));} \\ _Amiram Eldar_, Nov 08 2022
%Y Column 14 of A263950.
%Y Cf. A000010, A000203, A013671, A013672.
%K nonn,mult
%O 1,2
%A _N. J. A. Sloane_, Nov 19 2009
%E Definition corrected by _Enrique Pérez Herrero_, Oct 30 2010